Sparse Approximation Method Research Articles

Using a hyperbolic cross to solve non-linear macroeconomic models

The paper presents a sparse grid approximation method based on the hyperbolic cross and applies it to solve non-linear macroeconomic models. We show how the standard hyperbolic cross can be extended to give greater control over the approximating grid and we discuss how to implement an anisotropic hyperbolic cross. Applying the approximation method to four macroeconomic models, we establish that it delivers a level of accuracy on par or better than Smolyak's method and that it can produce accurate approximations using fewer points than Smolyak's method.

Sparse successive approximation for nonlinear [formula omitted] and [formula omitted] optimal control problems under residual errors

Successive approximation techniques are effective approaches to solve the Hamilton–Jacobi–Bellman (HJB)/Hamilton–Jacobi–Isaacs (HJI) equations in nonlinear H2 and H∞ optimal control problems (OCPs), but residual errors in the solving process may destroy its convergence property, and related numerical methods also pose computational burden and difficulties. In this paper, the HJB/HJI partial differential equations (PDEs) for infinite-horizon nonlinear H2 and H∞ OCPs are handled in a unified formulation, and a sparse successive approximation method is proposed. Taking advantage of successive approximation techniques, the nonlinear HJB/HJI PDEs are transformed into sequences of easily solvable linear PDEs, to which the solutions can be computed point-wise by handling simple initial value problems. Extra constraints are also incorporated in the solving process to guarantee the convergence under residual errors. The sparse grid based collocation points and basis functions are then employed to enable efficient numerical implementation. The performance of the proposed method is also numerically demonstrated in simulations.

Using Giant Pulses to Measure the Impulse Response of the Interstellar Medium

Giant pulses emitted by PSR B1937+21 are bright, intrinsically impulsive bursts. Thus, the observed signal from a giant pulse is a noisy but direct measurement of the impulse response from the ionized interstellar medium. We use this fact to detect 13,025 giant pulses directly in the baseband data of two observations of PSR B1937+21. Using the giant pulse signals, we model the time-varying impulse response with a sparse approximation method, in which the time dependence at each delay is decomposed in Fourier components, thus constructing a wavefield as a function of delay and differential Doppler shift. We find that the resulting wavefield has the expected parabolic shape with several diffuse structures within it, suggesting the presence of multiple scattering locations along the line of sight. We also detect an echo at a delay of about 2.4 ms, over 1.5 times the rotation period of the pulsar, which moves along the trajectory expected from geometry between the two observations. The structures in the wavefield are insufficiently sparse to produce a complete model of the system; hence, the model is not predictive across gaps larger than about the scintillation time. Nevertheless, within its range, it reproduces about 75% of the power of the impulse response, a fraction limited mostly by the signal-to-noise ratio of the observations. Furthermore, we show that by deconvolution, using the model impulse response, we can successfully recover the intrinsic pulsar emission from the observed signal.

Reconstructing Internal Tides Field Based on Sampling by Autonomous Underwater Vehicles

This article explores the challenges and solutions of using autonomous underwater vehicles (AUVs) to observe internal waves in the ocean. Internal waves are highly dynamic and complex, and their rapid spatial and temporal variability makes it difficult to accurately observe them using AUVs due to Doppler smearing and aliasing effects. To overcome these challenges, the article proposes a Sparse Approximation (SA) method that reconstructs the sectional field of internal tides using AUV sampling data on horizontal-reciprocating vertical-dimension sawtooth (i.e., yo-yo) trajectories. The SA method is validated by numerical simulations in a temperature field of internal tides, and a series of reconstructions comparing the SA and linear interpolation method are conducted. The results show that the SA method produces accurate results if the signal frequency of the field is less than the Nyquist frequency of the water column sampling near the horizontal sides of the section. Notably, it reveals that the Doppler smearing and aliasing effects in the SA reconstructions can be effectively suppressed in the vertical isotherm fluctuation frequency range which is less than the Nyquist frequency of the water column sampling near the horizontal sides of the section. Additionally, the article evaluates the robustness of the SA method under different conditions, including AUV sampling behaviors, temperature field variation, AUV sensing and navigation errors. The findings demonstrate that the SA reconstruction method can be effectively used to observe internal tides with AUVs, and provide a design principle for an AUV reciprocating yo-yo sampling trajectory to achieve feasible field reconstruction of internal tides.

Underwater ice adaptive mapping and reconstruction using autonomous underwater vehicles

The undersides of floating ice shelves and sea ice in the Antarctic and Arctic are among the least accessible environments on Earth. The interactions between ice shelves, sea ice, and the ocean are of considerable scientific interest. In order to fully understand the complex picture of sea ice, and not just its surface, it is quite necessary to map the underside to comprehend the full context of its growth and decay patterns. Autonomous Underwater Vehicles (AUVs) are rapidly becoming the desired platform of choice for mapping the underside of sea ice to provide high-resolution 3D views of sea ice topography. To increase the efficiency and accuracy of AUV sampling behaviors is significant for the under-ice observation mission given its limited endurance. In this paper, we present a low-cost underwater ice mapping framework for small-sized AUVs using adaptive sampling and map reconstruction methods. A small-sized AUV is cost-effective and convenient for operation in polar regions; however, due to its limited loading capacity and energy, it is more applicable for the vehicle to carry single-beam sonar for ice bottom mapping but not multi-beam. Thus, the essential issue in this application is how to obtain the key information of ice topography and how to reconstruct the map of ice draft (namely underwater ice thickness) with AUV sparse mapping swathes. To address this, we propose a graphics-based adaptive mapping method to densify the measuring of ice bottom surface with ‘noticeable’ variations; moreover, we also present a sparse approximation method for ice draft map reconstruction using the sparse mapping swathes from a single-beam sonar. Our efforts are to introduce an effective and efficient approach for underwater ice mapping using low-cost small-sized AUVs. Our proposed adaptive mapping and reconstruction methods are validated in the under-ice scenario created using the field data.

Data-Driven Time-Frequency Method and Its Application in Detection of Free Gas Beneath a Gas Hydrate Deposit

The time-frequency (TF) analysis method plays a significant role in the detection of natural gas hydrates. As a data-driven method, compressed sensing (CS) has been widely used in the TF methods due to the sparsity of the TF representation. This study proposes a data-driven TF method based on the CS theory and a non-convex regularization. In the implementation, a continuous wavelet transform (CWT) with a generalized beta wavelet (GBW) is formulated as an inverse problem based on the CS theory. The selection of appropriate parameters enables the GBW to match the seismic wavelets better than the widely used Morlet wavelet. The GBW can constitute a tight frame to reduce calculation time, particularly for large-scale field data processing. Additionally, the proposed TF method introduces the generalized minimax concave (GMC) penalty function as a non-convex regularization term. Compared with the classical sparse approximation method with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1} $ </tex-math></inline-formula> regularization, the GMC regularization term can enhance the sparsity in sparse inverse problems and ensure the convexity of sparse inversions. This article also presents an exponentially decreasing threshold scheme to adaptively select the regularization parameters. Three synthetic examples are investigated to demonstrate the performance of the proposed sparse TF representation with GMC regularization. Finally, the proposed TF method’s performance in detecting free gas of gas hydrates is validated using field seismic data obtained from the Blake Ridge.

Recovering the underlying trajectory from sparse and irregular longitudinal data

In this article, we consider the problem of recovering the underlying trajectory when the longitudinal data are sparsely and irregularly observed and noise‐contaminated. Such data are popularly analyzed with functional principal component analysis via the principal analysis by conditional estimation (PACE) method. The PACE method may sometimes be numerically unstable because it involves the inverse of the covariance matrix. We propose a sparse orthonormal approximation (SOAP) method as an alternative. It estimates the optimal empirical basis functions in the best approximation framework rather than eigen‐decomposing the covariance function. The SOAP method avoids estimating the mean and covariance function, which is challenging when the assembled time points with observations for all subjects are not sufficiently dense. The method also avoids the inverse of the covariance matrix, hence the computation is more stable. It does not require the functional principal component scores to follow the Gaussian distribution. We show that the SOAP estimate for the optimal empirical basis function is asymptotically consistent. The finite‐sample performance of the SOAP method is investigated in simulation studies in comparison with the PACE method. Our method is demonstrated by recovering the CD4 percentage curves from sparse and irregular data in a multi‐centre AIDS cohort study.

A New Adaptive Sparse Pseudospectral Approximation Method and its Application for Stochastic Power Flow

With the increasing integration of various renewable energy resources into electric power systems, the stochastic power flow calculation, which falls into the uncertainty propagation (UP) problems, has attracted renewed interest recently. The adaptive sparse pseudospectral approximation method (A-SPAM) is highly efficient to solve UP problems. However, there still exist two major drawbacks: 1) its global error estimate is not accurate enough to evaluate the actual approximation error; and 2) it is intended for single-output systems and there is no efficient method for its application in multi-output systems. In order to overcome these two defects, in this study a new adaptive sparse pseudospectral approximation method (NA-SPAM) is proposed based on two improvements on A-SPAM. First, a new global error estimate is presented which features much higher estimation accuracy. Second, a revised version of error estimates is given based on unification of the index set, which apparently improves the calculation efficiency for multi-output problems. At last, the effectiveness of the two improvements and NA-SPAM is validated by mathematical and practical cases.

Simultaneous‐source deblending using adaptive coherence‐constrained dictionary learning and sparse approximation

ABSTRACTThe dictionary learning and sparse approximation method using the K‐singular value decomposition algorithm rely on the knowledge of the sparsity or noise variance as a constraint when it is used for data denoising. However, the determination of the sparsity or noise variance of seismic data can be tricky and sometimes unknown, especially in seismic field data. Thus, where the cardinality or the noise variance is not known, the intrinsic character of the relative coherence between the removed noise from noisy data and its learned dictionary is instead used as a constraint for the sparse approximation of simultaneous‐source seismic data. The dictionary learning is obtained using a modified orthogonal matching pursuit algorithm which uses coherence as a constraint and is referred to as coherence dictionary learning. The coherence dictionary learning is then adapted to handle the simultaneous‐source seismic data deblending. A blending structure with random time dithering of sequential source shooting is used to guarantee adequate randomness of the noise. Two‐dimensional overlap patches of the noisy data were extracted from the common receiver gather domain to train the dictionary and to determine the sparse representation of the signal. The method is tested on both synthetic and field data, and it shows adequate data recovery. Comparing the result of this method to the matching pursuit algorithm constrained by the signal sparsity and the noise variance reveals that our approach performs better at noise attenuation and yields a reasonable data recovery especially for strong seismic signal.

A Sparse Nonstationary Trigonometric Gaussian Process Regression and Its Application on Nitrogen Oxide Prediction of the Diesel Engine

Gaussian process regression (GPR) has shown superiority in terms of state estimation for its nonparametric characteristic and uncertainty prediction ability. Due to its heavy computational complexity, GPR is generally used for small datasets. To efficiently deal with the big data, the sparse spectrum approximation method has been successfully applied to GPR to decrease the computational complexity. However, the stationarity of this method is a strict assumption for data and usually mismatches the industrial processes. In this article, we proposed a sparse nonstationary GPR, which can deal with the nonstationary relationship among samples and make the model more flexible, to settle the aforementioned problems. Furthermore, the performance of the proposed method is evaluated using three public datasets and a sampled diesel engine dataset, and the results show the superiority of our proposed method in terms of accuracy.

Feature-fusion-kernel-based Gaussian process model for probabilistic long-term load forecasting

In this paper, we present a feature fusion method designed for the Gaussian process model’s kernel functions for the probabilistic long-term load forecasting. To enrich the amount of information sent to the Gaussian kernel function, we propose a data conversion method that can convert one dimension time series data into multidimensional feature space. Besides, to improve computing efficiency in this data conversion method, we design a sparse approximation method according to the variational free energy principle with the Determinantal Point Processes (DPPs) as the pseudo inputs selection method. Also, the probabilistic distribution of our prediction is specially modified to bridge further the gap between the forecast and the original data set. Our approach is tested and presents a notable performance in the Global Energy Forecasting Competition data set.

A Compressed Sensing-Based Element Failure Diagnosis Method for Phased Array Antenna During Beam Steering

A novel diagnosis method for phased array antenna element failure using near-field measurements based on compressed sensing is proposed. In consideration of the beam steering property of phased arrays, sampling of the radiated field is performed with fixed probes. The independence of received field data is attained in the phase domain instead of the conventional spatial domain. It yields a new way to construct the sensing matrix and, hence, the measurement vector. This strategy enables onsite diagnosis while the array under test is still scanning regularly. The reconstruction of antenna element excitations is achieved via sparse approximation methods. A nonconvex optimization algorithm is utilized to reconstruct the unknown excitation amplitudes in consideration of noise. The detection area of a single probe is limited. Arrays consisting of isotropic antennas and Hertzian dipoles are tested to verify the effectiveness of the proposed method.

Bayesian hypothesis testing and experimental design for two-photon imaging data.

Variability, stochastic or otherwise, is a central feature of neural activity. Yet the means by which estimates of variation and uncertainty are derived from noisy observations of neural activity is often heuristic, with more weight given to numerical convenience than statistical rigour. For two-photon imaging data, composed of fundamentally probabilistic streams of photon detections, the problem is particularly acute. Here, we present a statistical pipeline for the inference and analysis of neural activity using Gaussian Process regression, applied to two-photon recordings of light-driven activity in ex vivo mouse retina. We demonstrate the flexibility and extensibility of these models, considering cases with non-stationary statistics, driven by complex parametric stimuli, in signal discrimination, hierarchical clustering and other inference tasks. Sparse approximation methods allow these models to be fitted rapidly, permitting them to actively guide the design of light stimulation in the midst of ongoing two-photon experiments.

Adjusted Sparse Tensor Product Spectral Galerkin Method for Solving Pseudodifferential Equations on the Sphere with Random Input Data

An adjusted sparse tensor product spectral Galerkin approximation method based on spherical harmonics is introduced and analyzed for solving pseudodifferential equations on the sphere with random input data. These equations arise from geodesy where the sphere is taken as a model of the earth. Numerical solutions to the corresponding $k$-th order statistical moment equations are found in adjusted sparse tensor approximation spaces which are accordingly designed to the regularity of the data and the equation. Established convergence theorem shows that the adjusted sparse tensor Galerkin discretization is superior not only to the full tensor product but also to the standard sparse tensor counterpart when the statistical moments of the data are of mixed unequal regularity. Numerical experiments illustrate our theoretical results.

Complex analysis of pre-seismic geoacoustic and electromagnetic emission signals

The article describes the results of complex analysis of pre-seismic signals of electromagnetic and geoacoustic radiation. We analyzed the frequency content of single sferics and geoacoustic impulses recorded before the Zhupanov earthquake that occurred on January 30, 2016. The signals were analyzed using sparse approximation method, in particular Adaptive Matched Pursuit. Background signals were studied together with pre-seismic ones. Distributions of frequencies, that are part of background and pre-seismic signals, were compared. Differences in the frequency content of pre-seismic sferics and geoacoustic impulses were found. The revealed features of pre-seismic signals in the future can be used in the design of systems for monitoring, forecasting and prevention of natural disasters. The research was supported by Russian Science Foundation (project No. 18-11-00087).

L1-norm Minimization based Sparse Approximation Method of EEG for Epileptic Seizure Detection

L1-norm Minimization based Sparse Approximation Method of EEG for Epileptic Seizure Detection

Children exposure to femtocell in indoor environments estimated by sparse low-rank tensor approximations

The exposure of an 8-year-old child to a femtocell operating at 2600 MHz, both (child and source) freely located in random positions in an indoor environment, was assessed. In order to develop surrogate models of the exposure, stochastic dosimetry based on sparse low-rank tensor approximation method (sparse LRA) was used. The surrogate models were used for fastly estimating the specific absorption rate (SAR) in all the possible positions of femtocell and child. Results showed that, for all the possible positions in the room, the exposure values were significantly below the International Commission of Non-Ionizing Radiation Protection (ICNIRP) guidelines for general public and that the probability of reaching SAR values higher than 1% of the ICNIRP guidelines value was lower than 0.006. The variation of the distance between femtocell and child influenced greatly the exposure, resulting in quartile coefficient of dispersion values always higher than 48%.

Bilinear Matrix Factorization Methods for Time-Varying Narrowband Channel Estimation: Exploiting Sparsity and Rank

In this paper, the estimation of a narrowband time-varying channel under the practical assumptions of finite block length and finite transmission bandwidth is investigated. It is shown that the signal, after passing through a time-varying narrowband channel reveals a useful parametric low-rank structure that can be represented as a bilinear form. To estimate the channel, two strategies are developed. The first method exploits the low-rank bilinear structure of the channel via a non-convex strategy based on an alternating direction optimization between the delay and Doppler directions. While prior Wirtinger flow methods have exhibited good performance with proper initialization, this is not true in the current scenario. Due to the non-convex nature of this approach, this first approach is sensitive to local minima. Furthermore, the convergence rate of the Wirtinger flow method is shown to be provably modest. Thus, a novel convex approach based on the minimization of the atomic norm using measurements of the signal at the time domain is proposed based on a second bilinear parametrization of the channel. For the convex approach, optimality and uniqueness conditions, and a theoretical guarantee for noiseless channel estimation with small number of measurements are characterized. Numerical results show that the performance of the proposed algorithm is independent of the leakage effect and the new methods can achieve a 5–12 dB improvement, on average, compared to a classical $l_1$ -based sparse approximation method that does not consider the leakage effect, and 2 dB improvement over a basis expansion method that considers the leakage effect.

A sparse approximation method for Ultrasonic Monitoring the degradation of adhesive joints

Ultrasonic pulse-echo methods have played a significant role in monitoring the integrity of adhesive joints along their service life. However, since for thin layers the echoes from two successive interfaces overlap in time, it is difficult to extract the parameters of the individual echoes. Over the last decade, sparse approximation methods have been extensively used to address this issue. These methods employ a large dictionary of elementary functions (atoms) and attempt to select the smallest subset of atoms (sparest approximation) that represent the ultrasonic signal accurately. In order to obtain highly sparse results, the dictionary should contain atoms which are similar to the anticipated ultrasonic echoes. However, when ultrasonically monitoring layered joints for long periods, it is expected that the ultrasonic echoes' shape will change significantly due to degradation of the layers. This will enforce the design of a large dictionary which includes all possible echo waveforms during the monitoring period. Using such a large dictionary may degrade the accuracy of the approximation and will result in high computational cost.In this paper we propose to apply a sparse approximation method, utilizing a small dictionary, just for signals measured before the degradation started. Then, we update the obtained atoms in order to find approximations for signals acquired after the degradation started. Our atom modification procedure is based on a physical model describing only the changes which occurred during the degradation period. The effectiveness of the proposed method is demonstrated through simulations and experiments involving environmentally degraded samples.

Thermal Tomography with Unknown Boundary

Thermal tomography is an imaging technique for deducing information about the internal structure of a physical body from temperature measurements on its boundary. This work considers time-dependent thermal tomography modeled by a parabolic initial/boundary value problem without accurate information on the exterior shape of the examined object. The adaptive sparse pseudospectral approximation method is used to form a polynomial surrogate for the dependence of the temperature measurements on the thermal conductivity, the heat capacity, the boundary heat transfer coefficient, and the body shape. These quantities can then be efficiently reconstructed via nonlinear, regularized least squares minimization employing the surrogate and its derivatives. The functionality of the resulting reconstruction algorithm is demonstrated by numerical experiments based on simulated data in two spatial dimensions.